Existence of a renormalized solution of a parabolic problem in anisotropic Sobolev–Orlicz spaces

We consider the first mixed problem for a certain class of anisotropic parabolic equations of the form
$$ (\beta(x,u))'_t-\operatorname{div} a(t,x,u,\nabla u) -b(t,x,u,\nabla u)=\mu $$
where $\mu$ is a measure and the coefficients contain noonpower nonlinearities in the cylindrical domain $D^T=(0,T)\times\Omega$, where $\Omega\subset \mathbb{R}^n$ is a bounded domain. We prove the existence of a renormalized solution of the problem for $g_t=0$ and a function $\beta(x,r)$, which increases with respect to $r$ and satisfies the Carathéodory condition.